Generalized Dirichlet and Neumann Boundary-Value Problems

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Space ◽  
Maximum Principle ◽  
Variational Inequalities ◽  
Eigenvalue Problems ◽  
Neumann Boundary ◽  
Neumann Problems ◽  
Weak Maximum Principle ◽  
General Elliptic ◽  
Neumann Boundary Value Problems ◽  
Generalized Dirichlet

In this chapter, the generalized or weak interpretation of the Dirichlet and Neumann problems for general elliptic expressions is motivated and then the Lax–Milgram Theorem is used to set the problems in the framework of eigenvalue problems for operators acting in Hilbert space. Results on variational inequalities in Chapter IV are applied to establish Stampacchia’s weak maximum principle, and this leads to the notion of capacity.

scholarly journals Asymptotic spectrum of multiparameter eigenvalue problems

1996 ◽  
Vol 39 (1) ◽  
pp. 119-132 ◽  
Author(s):  
Hans Volkmer
Keyword(s):  
Hilbert Space ◽  
Maximum Principle ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Asymptotic Spectrum ◽  
Multiparameter Eigenvalue Problem ◽  
Sturm Liouville

Results are given for the asymptotic spectrum of a multiparameter eigenvalue problem in Hilbert space. They are based on estimates for eigenvalues derived from the minim un-maximum principle. As an application, a multiparameter Sturm-Liouville problem is considered.

scholarly journals Perturbation of eigenvalues due to gaps in two-dimensional boundaries

2006 ◽  
Vol 463 (2079) ◽  
pp. 759-786 ◽  
Author(s):  
Anthony M.J Davis ◽  
Stefan G Llewellyn Smith
Keyword(s):  
Eigenvalue Problems ◽  
Numerical Solutions ◽  
Neumann Boundary ◽  
Diffusion Problem ◽  
Length Scale ◽  
Two Dimensional ◽  
Constructive Approach ◽  
Term Outcome ◽  
Long Term Outcome

Motivated by problems involving diffusion through small gaps, we revisit two-dimensional eigenvalue problems with localized perturbations to Neumann boundary conditions. We recover the known result that the gravest eigenvalue is O (|ln  ϵ | −1 ), where ϵ is the ratio of the size of the hole to the length-scale of the domain, and provide a simple and constructive approach for summing the inverse logarithm terms and obtaining further corrections. Comparisons with numerical solutions obtained for special geometries, both for the Dirichlet ‘patch problem’ where the perturbation to the boundary consists of a different boundary condition and for the gap problem, confirm that this approach is a simple way of obtaining an accurate value for the gravest eigenvalue and hence the long-term outcome of the underlying diffusion problem.

scholarly journals TWO-DIMENSIONAL HYBRIDS WITH MIXED BOUNDARY VALUE PROBLEMS

2016 ◽  
Vol 56 (3) ◽  
pp. 245
Author(s):  
Marzena Szajewska ◽  
Agnieszka Tereszkiewicz
Keyword(s):  
Boundary Value Problems ◽  
Special Functions ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Dirichlet Condition ◽  
Two Dimensional ◽  
Neumann Type ◽  
Neumann Boundary Value Problems ◽  
Real Euclidean Space

Boundary value problems are considered on a simplex <em>F</em> in the real Euclidean space R<sup>2</sup>. The recent discovery of new families of special functions, orthogonal on <em>F</em>, makes it possible to consider not only the Dirichlet or Neumann boundary value problems on <em>F</em>, but also the mixed boundary value problem which is a mixture of Dirichlet and Neumann type, ie. on some parts of the boundary of <em>F</em> a Dirichlet condition is fulfilled and on the other Neumann’s works.

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